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arXiv:1812.00671 · New Journal of Physics

Topological order on the Bloch sphere

Rotem Liss, Tal Mor, Román Orús

Published Mar 1, 2020• 0 citations
arXivHTMLPDF

Abstract

Abstract A Bloch sphere is the geometrical representation of an arbitrary two-dimensional Hilbert space. Possible classes of entanglement and separability for the pure and mixed states on the Bloch sphere were suggested by [Boyer et al 2017 PRA 95 032 308]. Here we construct a Bloch sphere for the Hilbert space spanned by one of the ground states of Kitaev’s toric code model and one of its closest product states. We prove that this sphere contains only one separable state, thus belonging to the fourth class suggested by the said paper. We furthermore study the topological order of the pure states on its surface and conclude that, according to conventional definitions, only one state (the toric code ground state) seems to present non-trivial topological order. We conjecture that most of the states on this Bloch sphere are neither ‘trivial’ states (namely, they cannot be generated from a product state using a trivial circuit) nor topologically ordered. In addition, we show that the whole setting can be understood in terms of Grover rotations with gauge symmetry, akin to the quantum search algorithm.

DOI
10.1088/1367-2630/ab7bf2

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Abstract A Bloch sphere is the geometrical representation of an arbitrary two-dimensional Hilbert space. Possible classes of entanglement and separability for the pure and mixed states on the Bloch sphere were suggested by [Boyer et al 2017 PRA 95 032 308]. Here we construct a Bloch sphere for the Hilbert space spanned by one of the ground states of Kitaev’s toric code model and one of its closest product states. We prove that this sphere contains only one separable state, thus belonging to the fourth class suggested by the said paper. We furthermore study the topological order of the pure states on its surface and conclude that, according to conventional definitions, only one state (the toric code ground state) seems to present non-trivial topological order. We conjecture that most of the states on this Bloch sphere are neither ‘trivial’ states (namely, they cannot be generated from a product state using a trivial circuit) nor topologically ordered. In addition, we show that the whole setting can be understood in terms of Grover rotations with gauge symmetry, akin to the quantum search algorithm.
Topological phases of matter offer a promising platform for quantum computation and quantum error correction. Nevertheless, unlike its counterpart in pure states, descriptions of topological order in mixed states remain relatively underexplored. Our work gives two definitions for replica topological order in mixed states, which involve nnn copies of density matrices of the mixed state. Our framework categorizes topological orders in mixed states as either quantum, classical, or trivial, depending on the type of information that can be encoded. For the case of the toric code model in the presence of decoherence, we associate for each phase a quantum channel and describes the structure of the code space. We show that in the quantum-topological phase, there exists a postselection-based error correction protocol that recovers the quantum information, while in the classical-topological phase, the quantum information has decohere and cannot be fully recovered. We accomplish this by describing the mixed state as a projected entangled pairs state (PEPS) and identifying the symmetry-protected topological order of its boundary state to the bulk topology. Using this formalism, we enumerate all the possible mixed state phases which result from applying a local decoherence channel to the toric code. In addition to the classical-topological phases that have been previously reported on, we also find mixed states exhibiting chiral topological order. We discuss the extent that our findings can be extrapolated to $n\ensuremath{\rightarrow}1$ limit.
Entanglement is an important concept in quantum information, quantum communication, and quantum computing. We provide a geometrical analysis of entanglement and separability for all the rank 2 quantum mixed states: complete analysis for the bipartite states and partial analysis for the multipartite states. For each rank 2 mixed state, we define its unique Bloch sphere, that is spanned by the eigenstates of its density matrix. We characterize those Bloch spheres into exactly five classes of entanglement and separability, give examples for each class, and prove that those are the only classes.
Abstract Toric Code, a cornerstone of the topological order, admits numerous realizations on different planar graphs. We prove that all these realizations share a single, universal entanglement structure in a sense that any Toric Code state can be generated by applying a fixed set of non-local unitaries between similar quasi-one-dimensional quantum ladders. In particular, we consider Toric Code states corresponding to various planar graphs and apply non-local dientanglers to qubits corresponding to non-contractible cycles that satisfy a topological constraint. We demonstrate that, independent of the geometry of the underlying graph, disentanglers convert Toric Code states into a tensor product of Kitaev's Ladder states. This exact and graph-independent construction reveals that the long-range entanglement characterizing the Toric Code's topological phase has a precise decomposable structure in a sense that it arises entirely from a specific pattern of entanglement between lower-dimensional subsystems. This structural insight opens new avenues for classification and simulation of topological orders.
We extend Bloch sphere formalism to pure two-qubit systems. Combining insights from Geometric Algebra and the analysis of entanglement in different conjugate bases we identify two Bloch sphere geometry that is suitable for representing maximally entangled states. It turns out that the relative direction of the coordinate axes of the two Bloch spheres may be used to describe the states. Moreover, the coordinate axes of one Bloch sphere should be rignt-handed and those of the other one should be left-handed. We describe and depict separable and maximally entangled states as well as entangling and non-entangling rotations. We also offer a graphical representation of the workings of a CNOT gate for different inputs. Finally, we provide a way to also represent partially entangled states and describe entanglement measures related to the surface area of the sphere enclosing the state representation.
Here we investigate the connection between topological order and the geometric entanglement, as measured by the logarithm of the overlap between a given state and its closest product state of blocks. We do this for a variety of topologically ordered systems such as the toric code, double semion, colour code and quantum double models. As happens for the entanglement entropy, we find that for sufficiently large block sizes the geometric entanglement is, up to possible sub-leading corrections, the sum of two contributions: a bulk contribution obeying a boundary law times the number of blocks and a contribution quantifying the underlying pattern of long-range entanglement of the topologically ordered state. This topological contribution is also present in the case of single-spin blocks in most cases, and constitutes an alternative characterization of topological order for these quantum states based on a multipartite entanglement measure. In particular, we see that the topological term for the two-dimensional colour code is twice as much as the one for the toric code, in accordance with recent renormalization group arguments (Bombin et al 2012 New J. Phys. 14 073048). Motivated by these results, we also derive a general formalism to obtain upper- and lower-bounds to the geometric entanglement of states with a non-Abelian group symmetry, and which we explicitly use to analyse quantum double models. Furthermore, we also provide an analysis of the robustness of the topological contribution in terms of renormalization and perturbation theory arguments, as well as a numerical estimation for small systems. Some of the results in this paper rely on the ability to disentangle single sites from the quantum state, which is always possible for the systems that we consider. Additionally we relate our results to the behaviour of the relative entropy of entanglement in topologically ordered systems, and discuss a number of numerical approaches based on tensor networks that could be employed to extract this topological contribution for large systems beyond exactly solvable models.
Decoherence is a major obstacle to the preparation of topological order in noisy intermediate-scale quantum devices. Here, we show that decoherence can also give rise to new types of topological order. Specifically, we construct concrete examples by proliferating fermionic anyons in the toric code via local quantum channels. The resulting mixed states retain long-range entanglement, which manifests in the nonzero topological entanglement negativity, though the topological quantum memory is destroyed by decoherence. By comparison with the gapless spin liquid in pure states, we show that the identified states represent a novel intrinsic mixed-state topological order, which has no counterpart in pure states. Through the lens of quantum anomalies of 1-form symmetries, we then provide general constructions of intrinsic mixed-state topological order and reveal the existence of nonbosonic deconfined anyons as another key feature of these novel phases. The extended meaning and characterization of deconfined excitations and their statistics in mixed states are clarified. Moreover, when these deconfined anyons have nontrivial braiding statistics, we prove that the mixed states cannot be prepared via finite-depth local quantum channels from any bipartite separable states. We further demonstrate our construction using the decohered Kitaev honeycomb model and the decohered double-semion model. In the latter case, a surprising scenario arises where decoherence gives rise to additional types of deconfined anyons.